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A255320 Expansion of chi(-x) * psi(x^3) * psi(x^48) in powers of x where chi(), psi() are Ramanujan theta functions.

%I A255320 #11 Feb 16 2025 08:33:25
%S A255320 1,-1,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,
%T A255320 -1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,-1,0,0,2,0,0,0,0,0,0,0,1,
%U A255320 -1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0
%N A255320 Expansion of chi(-x) * psi(x^3) * psi(x^48) in powers of x where chi(), psi() are Ramanujan theta functions.
%C A255320 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A255320 G. C. Greubel, <a href="/A255320/b255320.txt">Table of n, a(n) for n = 0..2500</a>
%H A255320 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A255320 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A255320 Expansion of q^(-19/3) * eta(q) * eta(q^6)^2 * eta(q^96)^2 / (eta(q^2) * eta(q^3) * eta(q^48)) in powers of q.
%F A255320 Euler transform of a period 96 sequence.
%F A255320 a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = a(16*n + 9) = a(16*n + 13) = 0.
%F A255320 -2 * a(n) = A227395(3*n + 19). a(8*n) = A255317(n). a(16*n + 1) = -A255318(n). a(16*n + 5) = -A255319(n).
%F A255320 a(n) = (-1)^n * A256574(n). - _Michael Somos_, Apr 24 2015
%e A255320 G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 - x^33 + x^40 + x^48 - x^49 + ...
%e A255320 G.f. = q^19 - q^22 - q^34 + q^43 + q^67 - q^82 - q^118 + q^139 + q^163 + ...
%t A255320 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[2, 0, x^(24)] / (4 x^(51/8)), {x, 0, n}];
%o A255320 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 * eta(x^96 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^48 + A)), n))};
%o A255320 (PARI) {a(n) = my(A, p, e); if( n<0 || n%8 == 2, 0, A = factor(3*n + 19); 1/2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, -(p+e==3), p%8 > 4, 1-e%2, e+1)))}; /* _Michael Somos_, Apr 24 2015 */
%Y A255320 Cf. A227395, A255317, A255318, A255319, A256574.
%K A255320 sign
%O A255320 0,57
%A A255320 _Michael Somos_, Feb 21 2015